hello difference between factions

Vasari starbase, I presume?  Those can be a little bit challenging since you really don't have an easy way to avoid them.  Against the other starbases, just don't bother getting in weapons range.  Set your units to "hold ground" behavior and then bombard the starbases from a distance.  Given enough time, even 6 or 7 squads of bombers will take down a fully upgraded starbase. 

Advent is currently the weakest faction of the three.  Their only strong rush-game opener is multiple Halcyon-class capital ships, their economy isn't as strong as the other factions, and they have to put together elaborate combos to be competitive.  On the bright side, once Advent does get a mature fleet operational, they leave TEC in their dust and become the second-strongest faction overall. 

Advent does have some very powerful combos you can employ, and in fact you need to employ if you want to match the other factions.  The most potent against TEC is the "battleball".  Use the Iconus Guardian with its shield projection ability and combine it with your Progenitor Mothership's shield regeneration ability.  This vastly increases your ability to absorb damage, and it's not unheardof for Advent to fight equally-sized TEC fleets and win without taking a single casualty.  You can further improve on this by using the Halcyon Carrier's energy amplification to boost damage and its telekinetic push to keep strike craft out of striking range.  You can also use the Rapture Battlecruiser's vertigo ability to further reduce enemy damage.  Unfortuantely, the battleball stops working well once Vasari have maxed out their phase missile upgrades, since at this point 30% of their damage passes through your shields anyways.

Vasari, as you have discovered, is the strongest faction overall.  Powerful from the moment you research Assailants to when you're annihilating your enemies with Kostura cannons.  TEC is arguably more powerful on an early-game rush, and can get a strong economy rolling faster as well, but they cap out in power at about the 3-lab level and the other factions just keep getting stronger and stronger past that point.  The most brutal part about Vasari is their phase missiles.  Even ignoring the prospect of killing an enemy unit without having to clear its shields first, phase missiles essentially work out to a 60% damage bonus due to ignoring normal damage mitigation.  On top of this, their 20% damage boost upgrade brings the total benefit to 90%.  I needn't say that this is obscene; Vasari deal ridiculous sums of damage in the late-game, and most of the units that matter use phase missiles.

Probably. While the Advent is probably the least popular faction online, against the AI it really doesn't matter what faction you pick. The Advent's main strength is its many synergies between ships (especially its capitals) like the battle ball (Progenitor/Guardian Combo) that makes a diverse Advent fleet greater than the sum of its parts. Their LRF (the once dominant Illuminator) is no longer a competitive damage dealer, so if you were relying on those that may have been part of your problem.

Are illuminators really that inferior when compared to the other long ranged frigates? They should still be able to crush enemy light frigates handily, and that's got to be good for keeping your Guardians and Carriers (the backbone of your fleet, imo) alive.

Illuminators suffer three major issues.  The first is that they require more labs to access, slowing down your rush or defense preparations.  The second is that their damage is spread out across three targets (and, if you're not a top-notch player, it's not even a sure-thing that all three weapons will be firing at any given time) which greatly reduces their effectiveness against large targets like capital ships, and also means the opponent can mitigate much of their damage by mixing damage sinks like flak frigates or hoshikos into their fleet.  Finally, their overall performance for cost just doesn't match up to what the Javelis or Assailant (with phase missiles) can offer.  All this comes together to make the illuminator a sub-par unit that just doesn't make the cut.

Mix some flaks with those light frigates to absorb damage, do some maneuvering to avoid the worst of the illuminators, and they will die to LRM/Assailants before they can kill light frigates.  It's kind of sad, but this is how it works.

These units just get run down by LRM/Assailants in the current version.  Advent really has nothing to match or blunt these high firepower units except large numbers of defense vessels, but such a fleet won't have enough illuminators to counter light frigates or damage output to threaten other unit types...

I still remember when they used to be the Uber unit of the game. Once someone had about 25 of them they won. Hands down. Turns out they were bugged.

Until Advent had a functioning battleball, TEC could still compete with LRM/Garda/Hoshiko if they played it well.  If Vasari could survive to get subverters and maxed out PM's, they also stood a chance later on.  However, Advent was rock-solid from the first minute to the last minute of the game, so it was uphill no matter who you were.